Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?

To prevent things from being too easy, I require two conditions on $R$:

1) $\operatorname{Spec}(R)$ should be connected (otherwise take $R = k_1 \times \ldots \times k_n$)

2) $R$ should be Noetherian (otherwise take $R$ to be a polynomial ring over $\mathbb{Z}$ in sufficiently many variables)

I would be happy with the case $n = 2$ (although I don't currently see how to get the general case from this). However, I do insist that the fields be arbitrary - it is known that any finite collection of *countable* fields is the set of residue fields of a PID (see this article by Heitmann - first page only).

(I've included the algebraic-geometry tag in hopes for some geometric insight. If however someone feels that this is sufficiently non-geometric, feel free to edit the tags.)